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Symmetry plays a basic role in variational problems (settled e.g. in $\mathbbR^{n}$ or in a more general manifold), for example to deal with the lack ofcompactness which naturally appear when the problem is invariant under theaction of a noncompact group. In $\mathbb R^n$, a compactness result forinvariant functions with respect to a subgroup $G$ of $\mathrm{O}(n)$ has beenproved under the condition that the $G$ action on $\mathbb R^n$ is compatible,see \cite{willem}. As a first result we generalize this and show here that thecompactness is recovered for particular subgroups of the isometry group of aRiemannian manifold. We investigate also isometric action on Hadamard manifold$(M,g)$ proving that a large class of subgroups of $\mathrm{Iso}(M,g)$ iscompatible. As an application we get a compactness result for ``invariant''functions which allows us to prove the existence of nonradial solutions for aclassical scalar equation and for a nonlocal fractional equation on $\mathbbR^n$ for $n=3$ and $n=5$, improving some results known in the literature.Finally, we prove the existence of nonradial invariant functions such that acompactness result holds for some symmetric spaces of non compact type.

A group theoretic proof of a compactness lemma and existence of nonradial solutions for semilinear elliptic equations